Chapter 6
Continuous Probability Distributions
Chapter 6
Continuous Probability Distributions
Case Problem: Specialty Toys
1.
Information provided by the forecaster
.025
.95
10,000
20,000
.025
30,000
At x = 30,000,
z
x

 

30, 000  20, 000

30, 000  20, 000
 5102
1.96
Normal distribution   20,000
2.
 1.96
  5102
@ 15,000
z
15, 000  20, 000
 0.98
5102
P(stockout) = 0.3365 + 0.5000 = 0.8365
@ 18,000
z
18, 000  20, 000
 0.39
5102
P(stockout) = 0.1517 + 0.5000 = 0.6517
@ 24,000
z
24, 000  20, 000
 0.78
5102
P(stockout) = 0.5000 - 0.2823 = 0.2177
CP - 24
© 2016 Cengage Learning. All Rights Reserved.
May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 6
Continuous Probability Distributions
@ 28,000
z
28, 000  20, 000
 1.57
5102
P(stockout) = 0.5000 - 0.4418 = 0.0582
3.
Profit projections for the order quantities under the 3 scenarios are computed below:
Order Quantity: 15,000
Sales
Unit Sales
10,000
20,000
30,000
Total Cost
240,000
240,000
240,000
at $24
240,000
360,000
360,000
at $5
25,000
0
0
Profit
25,000
120,000
120,000
Order Quantity: 18,000
Sales
Unit Sales
10,000
20,000
30,000
Total Cost
288,000
288,000
288,000
at $24
240,000
432,000
432,000
at $5
40,000
0
0
Profit
-8,000
144,000
144,000
Order Quantity: 24,000
Sales
Unit Sales
10,000
20,000
30,000
Total Cost
384,000
384,000
384,000
at $24
240,000
480,000
576,000
at $5
70,000
20,000
0
Profit
-74,000
116,000
192,000
Order Quantity: 28,000
Sales
Unit Sales
10,000
20,000
30,000
Total Cost
448,000
448,000
448,000
at $24
240,000
480,000
672,000
at $5
90,000
40,000
0
Profit
-118,000
72,000
224,000
CP - 25
© 2016 Cengage Learning. All Rights Reserved.
May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 6
4.
Continuous Probability Distributions
We need to find an order quantity that cuts off an area of .70 in the lower tail of the normal curve for
demand.
30%
70%
20,000 Q
z = 0.52
Q  20, 000
z
 0.52
5102
Q = 20,000 + 0.52(5102) = 22,653
The projected profits under the 3 scenarios are computed below.
Order Quantity: 22,653
Sales
Unit Sales
10,000
20,000
30,000
5.
Total Cost
362,488
362,488
362,488
at $24
240,000
480,000
543,672
at $5
63,265
13,265
0
Profit
-59,183
130,817
181,224
A variety of recommendations are possible. The students should justify their recommendation by
showing the projected profit obtained under the 3 scenarios used in parts 3 and 4. An order quantity
in the 18,000 to 20,000 range strikes a good compromise between the risk of a loss and generating
good profits.
While the students don't have the benefit of the following, a single-period inventory model
(sometimes called the news vendor model) shows how to find an optimal solution. We outline that
solution below.
A single-period inventory model recommends an order quantity that maximizes expected profit
based on the following formula:
P(Demand  Q* ) 
cu
cu  co
where P(Demand  Q* ) is the probability that demand is less than or equal to the recommended
order quantity, Q * . cu is the cost of underestimating demand (having lost sales because of a stockout)
and co is the cost per unit of overestimating demand (having unsold inventory). Specialty will sell
Weather Teddy for $24 per unit. The cost is $16 per unit. So, cu = $24 - $16 = $8. If inventory
remains after the holiday season, Specialty will sell all surplus inventory for $5 a unit. So, co = $16 $5 = $11.
CP - 26
© 2016 Cengage Learning. All Rights Reserved.
May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 6
Continuous Probability Distributions
P(Demand  Q* ) 
8
 0.4211
8  11
0.4211
0.5789
Q*
z = -0.20
z
Q  20,000
 0.20
5102
*
Q*  20, 000  0.20(5102)  18,980
The profit projections for this order quantity are computed below:
Order Quantity: 18,980
Sales
Unit Sales
10,000
20,000
30,000
Total Cost
303,680
303,680
303,680
at $24
240,000
455,520
455,520
at $5
44,900
0
0
Profit
-18,780
151,840
151,840
CP - 27
© 2016 Cengage Learning. All Rights Reserved.
May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.